Thucydides Roundtable, Book VI: The State with the Golden Arm

[by A. E. Clark]

T. Greer offers two metaphors for the restless dynamism which Alcibiades considered a necessity to the Athenian State in the summer of 415: a motorist’s climb up an icy hill, where if you do not keep moving forward you will slide back; and a child’s top, which must keep spinning or fall.

I believe these are good images for what Alcibiades wanted the Athenians to think.  Whether they are good images for the reality Athens faced (and needed to understand in order to make the right decision about Sicily) is another question.

A dynamic system can find equilibrium in a steady state. If enough angular energy continues to be imparted to the spinning top to compensate for the degrading effect of friction, that top will stand forever. A snapshot of the forces and resistances in play, if taken today, will be identical to the snapshot taken tomorrow or a year from now. The icy hillside is a little different, because it is hard to imagine the hill ascending forever. Apart from the geographic implausibility, as altitude increases both air pressure and temperature will fall, adding new difficulties to the vehicle’s operation.  But for a limited distance, assuming the gradient is constant and the ice uniform, it is likely that the motorist will find a steady-state solution: a constant speed that maintains traction up the hill.

Expanding empires encounter a complication that is absent from these examples.  By continuing to grow, they increase the burden of administration, the scale of required coordination, the potential for internal dissension, the number of things that can go wrong, and the vehemence of resistance to their reign. It is as if we said that the top must not only go on spinning but must carry a heavier weight with each passing hour; or that the car must ascend a hill that is becoming ever steeper. Reality enforces a limit on this kind of growth. In the parable of Icarus, closeness to the sun represents both the success of the enterprise and its catastrophic failure.

Dynamic systems in the social sciences are often modeled with mathematics. Such efforts require a great many variables whose values, as well as their partial first and even second derivatives, are linked in sprawling systems of equations. This science is a bit over my head, and I’m not sure it has ever proven notably successful in modeling social and economic realities. But there is a basic point central to this kind of math which many of us will remember from high-school physics: it is important to distinguish a value from its rate of change.  A car’s position is one thing (location); how that value changes with time is another thing (velocity); and how that rate of change itself changes with time is a third thing (acceleration). In this regard, Alcibiades shows some confusion:

… we have reached a position in which we must not be content with retaining what we have but must scheme to extend it for, if we cease to rule others, we shall be in danger of being ruled ourselves.” (6.18.3)

“If we cease to rule others, we shall be in danger of being ruled ourselves.” This is exactly what Pericles had said at 2.63.2: “to recede is no longer possible . . . to let [our empire] go is unsafe.”  Pericles’ rationale was as much psychological as economic: to restore freedom to any of those from whom Athens had taken it would make them all restless, with a cascading effect.  Assuming that the flow of tributary income Y is proportional to the stock of subject territory S, Pericles is warning that neither must diminish: dS/dt < 0 would spell danger for Athens. But mindful of the burdens, distractions, and risks of expanding the empire, he has also warned his countrymen “to attempt no new conquests” (2.65.7) for the duration of the war: dS/dt > 0 is also dangerous.

Yet Alcibiades’ conclusion is different: “We must not be content with retaining what we have but must scheme to extend it.”  It is not S, but dS/dt, that he would keep undiminished! In fact, considering the scale of the Sicilian expedition, Alcibiades was actually calling for dS/dt to rise, as success in Sicily would have not merely continued but accelerated the imperial expansion.

Alcibiades seems to err, therefore, because his conclusion does not follow from his Periclean premise; yet it is possible that his counsel, fatal though it was, rested on something other than a mathematical mistake.

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